Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T07:38:11.612Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite Element Computation of KPP Front Speeds in 3D Cellularand ABC Flows

Published online by Cambridge University Press:  12 June 2013

L. Shen ,
J. Xin  and
A. Zhou
L. Shen
Affiliation:
School of Mathematics, Capital Normal University, Beijing 100048, China
J. Xin*
Affiliation:
Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA
A. Zhou
Affiliation:
LSEC,Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
*
⋆ Corresponding author. E-mail: jxin@math.uci.edu
Get access

Abstract

We carried out a computational study of propagation speeds ofreaction-diffusion-advection fronts in three dimensional (3D) cellular andArnold-Beltrami-Childress (ABC) flows with Kolmogorov-Petrovsky-Piskunov(KPP)nonlinearity. The variational principle of front speeds reduces the problem to a principaleigenvalue calculation. An adaptive streamline diffusion finite element method is used inthe advection dominated regime. Numerical results showed that the front speeds areenhanced in cellular flows according to sublinear power lawO(δp),p ≈ 0.13, δ the flow intensity. In ABC flows however,the enhancement is O(δ) which can be attributed to thepresence of principal vortex tubes in the streamlines. Poincaré sections are used tovisualize and quantify the chaotic fractions of ABC flows in the phase space. The effectof chaotic streamlines of ABC flows on front speeds is studied by varying the threeparameters (a,b,c) of the ABC flows. Speed enhancement alongx direction is reduced as b (the parameter controlingthe flow variation along x) increases at fixed(a,c) > 0, more rapidly as the corresponding ABC streamlines becomemore chaotic.

Keywords

KPP fronts3D Cellular and ABC Flowseigenvalue problemsadaptive streamline diffusion finite element method
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 3: Front Propagation , 2013 , pp. 182 - 197
Copyright
© EDP Sciences, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abel, M., Cencini, M., Vergni, D., Vulpiani, A.. Front speed enhancement in cellular flows. Chaos, 12 (2002), pp. 481–488. CrossRefGoogle ScholarPubMed
Arnold, D.N., Mukherjee, A., Pouly, L.. Locally adapted tetrahedral meshes using bisection. SIAM J. Sci. Comput., 22 (2000), pp. 431–448. CrossRefGoogle Scholar
Audoly, B., Berestycki, H., Pomeau, Y.. Réaction diffusion en écoulement stationnaire rapide. C. R. Acad. Sci. Paris 328, Série IIb, 2000, pp. 255–262. Google Scholar
Babuška, I., Osborn, J.E.. Eigenvalue problems. in: Handbook of Numerical Analysis. Volume II, North-Holland, 1991, pp. 641–787.
Babuška, I., Vogelius, M.. Feedback and adaptive finite element solution of one-dimensional boundary value problems. Numer. Math., 44 (1984), pp. 75–102. CrossRefGoogle Scholar
Beattie, C.. Galerkin eigenvector approximations. Math. Comput., 69 (2000), pp. 1400–1434. CrossRefGoogle Scholar
Berestycki, H., Hamel, F.. Front propagation in periodic excitable media. Comm. Pure Appl. Math., 55 (2002), pp. 949–1032. CrossRefGoogle Scholar
Berestycki, H., Hamel, F., Nadirashvili, N.. Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena. Comm. Math. Physics, 253 (2005), pp 451–480. CrossRefGoogle Scholar
Biferale, L., Crisanti, A., Vergassola, M., Vulpiani, A.. Eddy diffusivities in scalar transport. Phys. Fluids, 7 (1995), pp. 2725–2734. CrossRefGoogle Scholar
Bourlioux, A., Khouider, B.. Rigorous asymptotic perspective on the large scale simulations of turbulent premixed flames. Multiscale Model. Simul., 6 (2007), pp. 287–307. CrossRefGoogle Scholar
Brandts, J., Krizek, M.. Gradient superconvergence on uniform simplicial partitions of polytopes. IMA J. Numer. Anal., 33 (2003), pp. 1–17. Google Scholar
Childress, S., Soward, A.M.. Scalar transport and alpha-effect for a family of cat’s eye flows. J. Fluid Mech, 205 (1989), pp. 99–133. CrossRefGoogle Scholar
Clavin, P., Williams, F.. Theory of premixed-flame propagation in large-scale turbulence. J. Fluid Mech., 90 (1979), pp. 598–604. CrossRefGoogle Scholar
Constantin, P., Kiselev, A., Oberman, A., Ryzhik, L.. Bulk burning rate in passive-reactive diffusion. Arch. Rat. Mech. Anal., 154 (2000), pp. 53–91. CrossRefGoogle Scholar
Codina, R. Comparison of some finite element methods for solving the diffusion-convection-reaction equation. Comput. Methods Appl. Mech. Engrg. 156 (1998), pp. 185–210. CrossRefGoogle Scholar
Dai, X., Xu, J., Zhou, A.. Convergence and optimal complexity of adaptive finite element eigenvalue computations. Numer. Math., 110 (2008), pp. 313–355. CrossRefGoogle Scholar
Dombre, T., Frisch, U., Greene, J.M., Hènon, M., Mehr, A., Soward, A.M.. Chaotic streamlines in the ABC flows, J. Fluid Mech., 67 (1986), pp. 353–391. CrossRefGoogle Scholar
E. Dormy, A. Soward, eds, “Mathematical Aspects of Natural Dynamics”, the Fluid Mech. of Astrophysics and Geophysics, Vol. 13, Grenoble Science and CRC Press, 2007.
Eriksson, K., Johnson, C.. Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems. Math. Comput., 60 (1993), pp. 167–188. CrossRefGoogle Scholar
Fannjiang, A., Papanicolaou, G.. Convection enhanced diffusion for periodic flows. SIAM J. Appl. Math., 54 (1992), pp. 333–408. CrossRefGoogle Scholar
Friedlander, S., Gilbert, A., Vishik, M.. Hydrodynamic instability for certain ABC flows. Geophys. Astrophys. Fluid Dynamics, 73 (1993), pp. 97–107. CrossRefGoogle Scholar
Friedlander, S., Vishik, M.. Dynamo theory, vorticity generation, exponential stretching. Chaos, 1(1991), pp. 198–205. CrossRefGoogle ScholarPubMed
T.J.R. Hughes, A.N. Brooks. A multidimensional upwind scheme with no crosswind diffusion. in: Finite Element Methods for Convection Dominated Flows (Hughes, T.J.R., ed.), New York, ASME, 1979.
Johnson, C., Nävert, U., Pitkäranta, J.. Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Engrg., 45 (1984), pp. 285–312. CrossRefGoogle Scholar
C. Johnson. Numerical Solution of Partial Differential Equations by the Finite Element Method, Combridge Univ. Press, Cambridge, 1987.
Majda, A., Souganidis, P.. Flame fronts in a turbulent combustion model with fractal velocity fields. Comm. Pure Appl. Math., 51 (1998), pp. 1337–1348. 3.0.CO;2-B>CrossRefGoogle Scholar
Mao, D., Shen, L., Zhou, A.. Adaptive finite element algorithms for eigenvalue problems based on local averaging type a posteriori error estimates. Adv. Comput. Math., 25 (2006), pp. 135–160. CrossRefGoogle Scholar
U. Nävert. A finite element method for convection-diffsion problems, PhD thesis, Chalmers University of Technology Göteberg, 1982.
Nolen, J., Xin, J.. Computing reactive front speeds in random flows by variational principle. Physica D, 237 (2008), pp. 3172–3177. CrossRefGoogle Scholar
Novikov, A., Ryzhik, L.. Boundary layers and KPP fronts in a cellular flow. Arch. Ration. Mech. Anal., 184 (2007), pp. 23–48. CrossRefGoogle Scholar
N. Peters, Turbulent Combustion, Cambridge University Press, Cambridge, 2000.
M. Proctor, A. Gilbert, eds, “Lectures on Solar and Planetary Dynamos”, Publications of the Newton Institute, Cambridge Univ Press, 1994.
P. Ronney. Some open issues in premixed turbulent combustion. in: Modeling in Combustion Science (J. D. Buckmaster and T. Takeno, Eds.), Lecture Notes in Physics, Vol. 449, Springer-Verlag, Berlin, pp. 3–22, 1995.
H-G Roos, M. Stynes, L. Tobiska. Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer series in computational mathematics, 24, second edition, 2008.
Ryzhik, L., Zlatos, A.. KPP pulsating front speed-up by flows. Comm. Math. Sci., 5 (2007), pp. 575–593. CrossRefGoogle Scholar
Shen, L., Xin, J., Zhou, A.. Finite element computation of KPP front speeds in random shear flows in cylinders. Multiscale Model. Simul., 7 (2008), pp. 1029–1041. CrossRefGoogle Scholar
Shen, L., Xin, J., Zhou, A.. Finite element computation of KPP front speeds in cellular and cat’s eye flows. J. Sci. Comput., 55(2), 2013, pp. 455-470. CrossRefGoogle Scholar
Shen, L., Zhou, A.. A defect correction scheme for finite element eigenvalues with applications to quantum chemistry. SIAM J. Sci. Comput., 28 (2006), pp. 321–338 CrossRefGoogle Scholar
Sivashinsky, G.. Cascade-renormalization theory of turbulent flame speed. Combust. Sci. Tech., 62 (1988), pp. 77–96. CrossRefGoogle Scholar
R. Verfüth. A Review of a Posteriori Error Estimates and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, New York, 1996.
Verfüth, R.. Robust a posteriori error estimates for stationary convection-diffusion equations. SIAM J. Numer. Anal., 43 (2005), pp. 1766–1782. CrossRefGoogle Scholar
F. Williams, Turbulent combustion. in: The Mathematics of Combustion (J. Buckmaster, ed.), SIAM, Philadelphia, pp. 97–131, 1985.
J. Xin. An Introduction to Fronts in Random Media, Surveys and Tutorials in the Applied Mathematical Sciences, Vol. 5, Springer, 2009.
J. Xin, Y. Yu. Analysis and comparison of large time front speeds in turbulent combustion models. http://arxiv.org/submit/255369, 2011.
Yakhot, V.. Propagation velocity of premixed turbulent flames. Comb. Sci. Tech., 60 (1988), pp. 191–214. CrossRefGoogle Scholar
Zlatos, A.. Sharp asymptotics for KPP pulsating front speed-up and diffusion enhancement by flows. Arch Rat. Mech. Anal, 195 (2010), pp.441–453. CrossRefGoogle Scholar
Zlatos, A.. Reaction-diffusion front speed enhancement by flows. Ann. Inst. H. Poincaré, Anal. Non Linaire, 28 (2011), pp. 711–726. CrossRefGoogle Scholar